##### URI
https://hdl.handle.net/10355/61884
 dc.contributor.advisor Banks, Wiliam eng dc.contributor.author Guo, Victor Zhenyu eng dc.date.issued 2017 eng dc.date.submitted 2017 Spring eng dc.description.abstract [ACCESS RESTRICTED TO THE UNIVERSITY OF MISSOURI AT AUTHOR'S REQUEST.] This thesis is focus on the methods of exponential sums and sieve methods applying to distribution of primes numbers in several forms, such as Piatetski-Shapiro primes, Beatty sequences, almost primes and primes in arithmetic progression. In the end, we also think about the classical problem in Burgess bound. We begin by explaining the importance of the methods of exponential sums. Together with sieve methods, we investigate the Piatetski-Shapiro primes from almost primes and the intersection between Piatetski-Shapiro primes and Betty sequences. Above all, we study primes in several forms from a "thin" integer set. We also study the distribution of consecutive prime numbers from two Beatty sequences by an assumption of a well-known conjecture. Finally, we turn to the methods of character sums and the problem of the least quadratic nonresidue. We improve the best known bound by changing the arbitrary small constant into a reciprocal of an infinite function. Possible future work is also discussed in the thesis. eng dc.identifier.uri https://hdl.handle.net/10355/61884 dc.language English eng dc.publisher University of Missouri--Columbia eng dc.relation.ispartofcollection University of Missouri--Columbia. Graduate School. Theses and Dissertations eng dc.rights Access to files is limited to the University of Missouri--Columbia. eng dc.subject.FAST Numbers, Prime -- Study and teaching eng dc.subject.FAST Sequences (Mathematics) eng dc.title Exponential sums, character sums, sieve methods and distribution of prime numbers eng dc.type Thesis eng thesis.degree.discipline Mathematics (MU) eng thesis.degree.level Doctoral eng thesis.degree.name Ph. D. eng
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